Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Finding the Vertex of a Parabolic Equation**
This exercise involves determining the vertex of the quadratic equation:
\[ y = -0.28x^2 - 1.0x + 2.1 \]
### Steps to Find the Vertex:
1. **Identify the coefficients**:
- \( a = -0.28 \)
- \( b = -1.0 \)
- \( c = 2.1 \)
2. **Use the vertex formula**:
The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
3. **Calculate the x-coordinate**:
\[ x = -\frac{-1.0}{2 \times -0.28} = -\frac{1.0}{-0.56} = 1.79 \]
4. **Calculate the y-coordinate**:
Substitute \( x = 1.79 \) back into the equation to find the y-coordinate:
\[ y = -0.28(1.79)^2 - 1.0(1.79) + 2.1 \]
\[ y \approx -0.28(3.2041) - 1.79 + 2.1 \]
\[ y \approx -0.897 + 2.1 - 1.79 \]
\[ y \approx -0.587 \]
Thus, the vertex of the parabola \( y = -0.28x^2 - 1.0x + 2.1 \) is approximately at \( (1.79, -0.587) \).
### Explanation:
The vertex represents the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. In this case, since \( a = -0.28 \) is negative, the parabola opens downwards indicating the vertex is the maximum point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d8be728-4e79-4510-aca0-fa71266ad769%2F4d2028b3-3238-4871-ac43-ba74be3c05cc%2F4ezt42g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Vertex of a Parabolic Equation**
This exercise involves determining the vertex of the quadratic equation:
\[ y = -0.28x^2 - 1.0x + 2.1 \]
### Steps to Find the Vertex:
1. **Identify the coefficients**:
- \( a = -0.28 \)
- \( b = -1.0 \)
- \( c = 2.1 \)
2. **Use the vertex formula**:
The x-coordinate of the vertex can be found using the formula:
\[ x = -\frac{b}{2a} \]
3. **Calculate the x-coordinate**:
\[ x = -\frac{-1.0}{2 \times -0.28} = -\frac{1.0}{-0.56} = 1.79 \]
4. **Calculate the y-coordinate**:
Substitute \( x = 1.79 \) back into the equation to find the y-coordinate:
\[ y = -0.28(1.79)^2 - 1.0(1.79) + 2.1 \]
\[ y \approx -0.28(3.2041) - 1.79 + 2.1 \]
\[ y \approx -0.897 + 2.1 - 1.79 \]
\[ y \approx -0.587 \]
Thus, the vertex of the parabola \( y = -0.28x^2 - 1.0x + 2.1 \) is approximately at \( (1.79, -0.587) \).
### Explanation:
The vertex represents the highest or lowest point of the parabola, depending on whether it opens upwards or downwards. In this case, since \( a = -0.28 \) is negative, the parabola opens downwards indicating the vertex is the maximum point.
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